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Mirrors > Home > MPE Home > Th. List > negne0i | Structured version Visualization version GIF version |
Description: The negative of a nonzero number is nonzero. (Contributed by NM, 30-Jul-2004.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
negne0i.2 | ⊢ 𝐴 ≠ 0 |
Ref | Expression |
---|---|
negne0i | ⊢ -𝐴 ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negne0i.2 | . 2 ⊢ 𝐴 ≠ 0 | |
2 | negidi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | 2 | negne0bi 10812 | . 2 ⊢ (𝐴 ≠ 0 ↔ -𝐴 ≠ 0) |
4 | 1, 3 | mpbi 231 | 1 ⊢ -𝐴 ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2081 ≠ wne 2984 ℂcc 10386 0cc0 10388 -cneg 10723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-po 5367 df-so 5368 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-ltxr 10531 df-sub 10724 df-neg 10725 |
This theorem is referenced by: neg1ne0 11606 iblcnlem1 24076 itgcnlem 24078 dvsincos 24266 tanregt0 24809 tanatan 25183 atantayl2 25202 |
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